Question

A pendulum has an angular amplitude $\theta $ . The tension in the string at extreme position is T₁ and at the bottom is T₂. If it is known that T₂ = 2T₁, then what is the value of $cos\theta $ ?

Answer 1

$\text{T}_{1} = \text{mg cos} \theta $ ..... (1)
$\mathrm{T}_2-\mathrm{mg}=\frac{\mathrm{mv}^2}{l}=\frac{\mathrm{m}}{l}(2 \mathrm{gh})$
$= \frac{2 \text{mg}}{\textit{l}} \left(1 - \text{cos} \theta \right) \textit{l}$
$= 2 \text{mg} \left(1 - \text{cos} \theta \right) \textit{l}$
or $\mathrm{T}_2=\mathrm{mg}+2 \mathrm{mg}(1-\cos \theta)$ ....... (2)

Given T₂ = 2T₁
or $\text{mg} + 2 \text{mg} \left(1 - \text{cos} \theta \right) = \text{2mg cos} \theta $
or $\text{cos} \theta = \frac{3}{4}$
= 0.75